In algebraic geometry, standard monomial theory describes the sections of a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...

over a generalized flag variety or

Schubert variety In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, \mathbf_k(V) of k-dimensional subspaces of a vector space V, usually with singular points. Like the Grassmannian, it is a kind of moduli space, whose elements sati ...

of a

reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation that has a finite kernel and is a ...

by giving an explicit basis of elements called standard monomials. Many of the results have been extended to

Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...

s and their groups. There are monographs on standard monomial theory by and and survey articles by and . One of important open problems is to give a completely geometric construction of the theory.M. Brion and V. Lakshmibai : A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680.

History

introduced monomials associated to standard

Young tableaux In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups ...

. (see also ) used Young's monomials, which he called standard power products, named after standard tableaux, to give a basis for the homogeneous coordinate rings of complex

Grassmannian In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...

s. initiated a program, called standard monomial theory, to extend Hodge's work to varieties ''G''/''P'', for ''P'' any

parabolic subgroup Parabolic subgroup may refer to: * a parabolic subgroup of a reflection group * a subgroup of an algebraic group that contains a Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zarisk ...

of any

in any characteristic, by giving explicit bases using standard monomials for sections of line bundles over these varieties. The case of Grassmannians studied by Hodge corresponds to the case when ''G'' is a special linear group in characteristic 0 and ''P'' is a maximal parabolic subgroup. Seshadri was soon joined in this effort by V. Lakshmibai and Chitikila Musili. They worked out standard monomial theory first for minuscule representations of ''G'' and then for groups ''G'' of classical type, and formulated several conjectures describing it for more general cases. proved their conjectures using the Littelmann path model, in particular giving a uniform description of standard monomials for all reductive groups. and and give detailed descriptions of the early development of standard monomial theory.

Applications

*Since the sections of line bundles over generalized flag varieties tend to form irreducible representations of the corresponding algebraic groups, having an explicit basis of standard monomials allows one to give character formulas for these representations. Similarly one gets character formulas for Demazure modules. The explicit bases given by standard monomial theory are closely related to crystal bases and Littelmann path models of representations. *Standard monomial theory allows one to describe the singularities of Schubert varieties, and in particular sometimes proves that Schubert varieties are normal or Cohen–Macaulay. *Standard monomial theory can be used to prove Demazure's conjecture. *Standard monomial theory proves the Kempf vanishing theorem and other vanishing theorems for the higher cohomology of effective line bundles over Schubert varieties. *Standard monomial theory gives explicit bases for some rings of invariants in

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...

. *Standard monomial theory gives generalizations of the

Littlewood–Richardson rule In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural number ...

about decompositions of tensor products of representations to all reductive algebraic groups. *Standard monomial theory can be used to prove the existence of good filtrations on some representations of reductive algebraic groups in positive characteristic.

Notes

References

* * * * * * * * * * * * *{{Citation , last1=Young , first1=Alfred , author1-link=Alfred Young (mathematician) , title=On Quantitative Substitutional Analysis , doi=10.1112/plms/s2-28.1.255 , year=1928 , journal= Proc. London Math. Soc. , volume=28 , issue=1 , pages=255–292, url=https://zenodo.org/record/1447746 Algebraic geometry Invariant theory